How to limit movement of a IK-chain (armature) to one plane?
The IK constraint generally ignores other constraints, so using limit constraints won't work.

However each bone has its own IK settings on their properties panel that allow you to limit rotations etc, for more information see https://docs.blender.org/manual/en/dev/rigging/armatures/posing/bone_constraints/inverse_kinematics/introduction.html#bone-ik-panel